Question

How do you find the equation of the straight line joining: $\left( {3, - 1} \right)$, $\left( {5,4} \right)$ ?

Answer 1

Here in this question, we have to find the equation of the straight line passing through the two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$. Find the equation by using the Point-Slope formula $y - {y_1} = m\left( {x - {x_1}} \right)$ before finding the equation first we have to find the slope using the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$. On simplification to the point-slope formula we get the required solution.

Complete step by step solution:
The general equation of a straight line is $y = mx + c$, where $m$ is the gradient or slope and $\left( {0,c} \right)$ the coordinates of the y-intercept. Consider, the point-slope formula,
$y - {y_1} = m\left( {x - {x_1}} \right)$-------(1)
The point-slope formula uses the slope and the coordinates of a point along the line to find the y-intercept.
Find the slope $m$ in point-slope formula by using the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Where ${x_1} = 3$, ${x_2} = 5$, ${y_1} = - 1$ and ${y_2} = 4$ on substituting this in formula, then
$m = \dfrac{{4 - \left( { - 1} \right)}}{{5 - 3}}$
$\Rightarrow \,\,\,m = \dfrac{{4 + 1}}{{5 - 3}}$
On simplification, we get
$m = \dfrac{5}{2}$
Now we get the gradient or slope of the line which passes through the points $\left( {3, - 1} \right)$ and $\left( {5,4} \right)$.
Substitute the slope m and the point $\left( {{x_1},{y_1}} \right) = \left( {3, - 1} \right)$ in the point slope formula.
Consider the equation (1)
$y - {y_1} = m\left( {x - {x_1}} \right)$
Where $m = \dfrac{5}{2}$, ${x_1} = 3$ and ${y_1} = - 1$ on substitution, we get
$y - \left( { - 1} \right) = \dfrac{5}{2}\left( {x - 3} \right)$
$\Rightarrow \,\,y + 1 = \dfrac{5}{2}x - \dfrac{5}{2}\left( 3 \right)$
$\Rightarrow \,\,y + 1 = \dfrac{5}{2}x - \dfrac{{15}}{2}$
Subtract 1 on both side, then
$y + 1 - 1 = \dfrac{5}{2}x - \dfrac{{15}}{2} - 1$
On simplification, we get
$y = \dfrac{5}{2}x - \left( {\dfrac{{15 + 2}}{2}} \right)$
$\Rightarrow \,\,y = \dfrac{5}{2}x - \dfrac{{17}}{2}$
Or it can be written as
$\therefore\,\,y = \dfrac{{5x - 17}}{2}$

Hence, the equation of the line passing through points $\left( {3, - 1} \right)$ and $\left( {5,4} \right)$ is $y = \dfrac{{5x - 17}}{2}$.

Note: The slope of a line is a ratio of the change in the y value and the change in the x value. We have to know the equation of a line and then we have to substitute the values to the equation, hence we can determine the value. While simplifying the equation we must take care of signs of terms.